Prime Factors | Common Divisors | Common Multiples | Divisors | Radical

Find Divisors of a Number

Calculator for the divisors or factors of an integer. The divisors are those integers by which the given number can be divided without a remainder. Every integer has a divisor of 1, the even numbers have a divisor of 2. If the sum of all divisors without themselves equals the number itself, it is a perfect number, the first perfect numbers are 6 and 28.

Integer:



Divisors:
Sum of the divisors without the number itself:
Sum of the divisors with the number itself:

The set of all divisors of a number n is denoted as T(n). For a prime number p, T(p) = {1, p}, since prime numbers are only divisible by 1 and themselves. Numbers with more than two divisors are called composite. A central result of elementary number theory is the fundamental theorem of arithmetic: Every natural number greater than 1 can be uniquely expressed as a product of prime numbers. This prime factorization allows the systematic determination of all divisors of a number. If n = p₁^a₁ * p₂^a₂ * ... * pk^ak is the prime factorization of n, then the number of divisors of n is given by (a₁+1)(a₂+1)...(ak+1). The sum of all divisors of a number n, including n itself, is denoted by σ(n). A number n is called perfect if σ(n) = 2n. The smallest perfect numbers are 6 and 28, since 1+2+3=6 and 1+2+4+7+14=28. Perfect numbers are closely related to Mersenne primes: Every even perfect number has the form 2^(p-1)(2^p-1), where 2^p-1 is a Mersenne prime.

Divisibility rules simplify checking whether one number is divisible by another without having to explicitly perform the division. A number is divisible by 2 if its last digit is even, by 3 if the sum of its digits is divisible by 3, and by 5 if its last digit is 0 or 5. These rules are based on the properties of the positional number system and modular arithmetic. They are used in cryptography, error detection in data transmissions and in the efficient implementation of algorithms.

Last updated on 04/08/2026. Author: Jürgen Kummer

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